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<p>We study the incompressible Euler equation and prove that the set of weak solutions is path-connected. More precisely, we construct paths of Hölder regularity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^{1/2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, valued in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^0_{t, loc} L^2_x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>C</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>l</mi> <mi>o</mi> <mi>c</mi> </mrow> <mn>0</mn> </msubsup> <msubsup> <mi>L</mi> <mi>x</mi> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> endowed with the strong topology. The main result relies on a convex integration construction adapted from the seminal work of De Lellis and Székelyhidi (Ann of Math (2) 170:1417-1436, 2009), extending it to a more broader geometric framework, replacing balls with arbitrary convex compact sets.</p>

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Path-connectedness of incompressible Euler solutions

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  • 发表日期：

    2026年1月19日

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  • Philippe Anjolras

摘要

We study the incompressible Euler equation and prove that the set of weak solutions is path-connected. More precisely, we construct paths of Hölder regularity \(C^{1/2}\) C 1 / 2 , valued in \(C^0_{t, loc} L^2_x\) C t , l o c 0 L x 2 endowed with the strong topology. The main result relies on a convex integration construction adapted from the seminal work of De Lellis and Székelyhidi (Ann of Math (2) 170:1417-1436, 2009), extending it to a more broader geometric framework, replacing balls with arbitrary convex compact sets.